By
Gödel's
incompleteness theorem, for any
reasonable (and
reasonably strong) theory there
exists an
undecidable sentence, i.e. a
statement in the
language of the theory which the
theory can
neither prove nor disprove.
Gödel's undecidable
sentences concerned logical
phenomena such as
provability and consistency, but
it gradually
became clear that undecidability
applies
also to statements with a
more evident
mathematical meaning.
For typical theories that
attempt to axiomatize "all of
mathematics", such as
Zermelo-Fraenkel set theory,
known undecidable statements
tend to concern matters rather
distant from the everyday
experience of most
mathematicians: the continuum
hypothesis, the existence of
very large cardinal numbers, and
so on. Eventually logicians
began to ask what sort of axioms
are needed to prove basic
theorems about more mundane
mathematical objects such as
e.g. natural and real numbers,
continuous functions or
separable metric spaces. I will
talk about a research programme
known as reverse mathematics,
which was initiated almost 50
years in an attempt to answer
such questions.